No full-text available
To read the full-text of this research,
you can request a copy directly from the author.
Global Actions and Related Objects Foundations
... The motivation for the introduction of global actions by A. Bak [2,3,4] was to provide an algebraic setting in which to bring higher algebraic K-theory nearer to the intuitions of the original work of J.H.C. Whitehead on K 1 (R). In this work, elementary matrices, and sequences of their actions on the general linear group, play a key rôle. ...
... Bak, in [2,3,4], for instance, found that to deal with stable and unstable higher K-groups it was useful to consider a family of subgroups of the elementary matrix group, and that the theory in general could be organised as a family of group actions, indexed by a set with a relation , often a partial order, and with certain 'patching conditions'. This became his 'global action', which he viewed as a kind of 'algebraic manifold', analogous to the notion of topological manifold, but in an algebraic setting. ...
... Intuitively a path in a global action A is a sequence of points a 0 , · · · , a n in |A| so that each a i , a i+1 , i = 0, · · · , n − 1 is a α-frame for some (varying) α ∈ Φ A . This idea can be captured using a morphism from a global action model of a line, and this is done in the initial papers on global actions, [2,3,4]. Here we postpone this until the third section as there is a certain technical advantage in considering the line with a groupoid atlas structure and that will be introduced there. ...
Global actions were introduced by A. Bak to give a combi-natorial approach to higher K-theory, in which control is kept of the elementary operations through paths and paths of paths. This paper is intended as an introduction to this circle of ideas, including the homotopy theory of global actions, which one obtains naturally from the notion of path of elementary operations. Emphasis is placed on developing examples taken from combinatorial group theory, as well as K-theory. The concept of groupoid atlas plays a clarifying role. dedicated to the memory of Saunders MacLane (1909-2005).
... The motivation for the introduction of global actions by A. Bak [2, 3, 4] was to provide an algebraic setting in which to bring higher algebraic K-theory nearer to the intuitions of the original work of J.H.C. Whitehead on K 1 (R). In this work, elementary matrices, and sequences of their actions on the general linear group, play a key rôle. ...
... This was a great result, but the excursion into topology has meant that the original combinatorial intuitions get somewhat lost. Bak, in [2, 3, 4], for instance, found that to deal with stable and unstable higher K-groups it was useful to consider a family of subgroups of the elementary matrix group, and that the theory in general could be organised as a family of group actions, indexed by a set with a relation , often a partial order, and with certain 'patching conditions'. This became his 'global action', which he viewed as a kind of 'algebraic manifold', analogous to the notion of topological manifold, but in an algebraic setting. ...
... Intuitively a path in a global action A is a sequence of points a 0 , · · · , a n in |A| so that each a i , a i+1 , i = 0, · · · , n − 1 is a α-frame for some (varying) α ∈ Φ A . This idea can be captured using a morphism from a global action model of a line, and this is done in the initial papers on global actions, [2, 3, 4]. Here we postpone this until the third section as there is a certain technical advantage in considering the line with a groupoid atlas structure and that will be introduced there. ...
Global actions were introduced by A. Bak to give a combi-natorial approach to higher K-theory, in which control is kept of the elementary operations through paths and paths of paths. This paper is intended as an introduction to this circle of ideas, including the homotopy theory of global actions, which one ob-tains naturally from the notion of path of elementary opera-tions. Emphasis is placed on developing examples taken from combinatorial group theory, as well as K-theory. The concept of groupoid atlas plays a clarifying role. dedicated to the memory of Saunders MacLane(1909-2005).
A global action is an algebraic analogue of a topological space. It consists
of group actions , , which
fulfill a certain compatibility condition. We investigate the homotopy theory
of global actions. The main result establishes a Galois type correspondence
between connected coverings of a given connected global action and subgroups of
the fundamental group of that action.
ResearchGate has not been able to resolve any references for this publication.